(c) Linearly independent: as
2
2
7
1

2
3
3

5
9
row reduces to
1
0
0
0
1
0
0
0
1
.
(d) Linearly independent: as
A
=
1
1
1
. . .
1
0
2
2
. . .
2
0
0
3
· · ·
3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0
0
. . .
n
row reduces to the identity
n
×
n
matrix.
11
7.
(a) Use GaussJordan elimination to solve the homogeneous system
x
1

x
2

4
x
3
=
0
x
1
+
x
2
+ 4
x
3
=
0
2
x
2
+
x
3
=
0
(b) Find by inspection a particular solution of the system
x
1

x
2

4
x
3
=

4
x
1
+
x
2
+ 4
x
3
=
6
2
x
2
+
x
3
=
3
and hence write down the general solution, using (a).
Solution
(a) We row reduce the corresponding augmented matrix:
1

1

4
0
1
1
4
0
0
2
1
0
1

1

4
0
0
2
8
0
0
2
1
0
1
0
0
0
0
1
4
0
0
0

7
0
1
0
0
0
0
1
0
0
0
0
1
0
which shows that there is a unique solution
x
1
x
2
x
3
=
0
0
0
.
(b) By inspection we see that substituting in
x
1
=
x
2
=
x
3
= 1 gives the desired result, hence
the general solution is
x
1
x
2
x
3
=
0
0
0
+
1
1
1
=
1
1
1
.
8. Let
A
=
1
5
2

1
9
5
1
3
1
and let
W
be the set of all linear combinations of the columns of
A
.
(a) Show that (4
,
10
,
2) is in
W
(you should be able to do this by inspection, that is, without
any row operations).
(b) Solve the homogeneous system with augmented matrix [
A

0
].
(c) Given that (1
,
2
,

1) is a solution of the linear system [
A

b
], what is the vector
b
?
(d) Write down the general solution of [
A

b
] (with the vector
b
from part (c)), without directly
solving this system.
Solution
(a) By inspection we see that
4
10
2
is given by the following linear combination of
the columns: 0
1

1
1
+ 0
5
9
3
+ 2
2
5
1
.
12
(b) Row reducing gives:
1
5
2
0

1
9
5
0
1
3
1
0
1
5
2
0
0
14
7
0
0

2

1
0
1
5
2
0
0
1
1
2
0
0
0
0
0
1
0

1
2
0
0
1
1
2
0
0
0
0
0
Let
x
3
=
t
, then
x
1
=
1
2
t
and
x
2
=

1
2
t
, so (
x
1
, x
2
, x
3
) =
t
1
2

1
2
1
, for any
t
∈
R
.
(c) The first coordinate of
b
is 1
×
1 + 5
×
2 + 2
×
(

1) = 9.
The second coordinate of
b
is

1
×
1 + 9
×
2 + 5
×
(

1) = 12. The third coordinate of
b
is 1
×
1 + 3
×
2 + 1
×
(

1) = 6. So
b
=
9
12
6
.
(d) The general solution of [
A

b
] is given by adding together the general solution of [
A

0
] and a
particular solution of [
A

b
]. In this case we have
x
=
1
2

1
+
t
1
2

1
2
1
,
t
∈
R
.
9. For which value(s) of
d
are the following sets of vectors linearly dependent? Justify your answers.
(a)
{
(1
,

1
,
4))
,
(3
,

5
,
5)
,
(

1
,
5
, d
)
}
(b)
{
(3
,
7
,

2)
,
(

6
, d,
4)
,
(9
,
1
,

4)
}
Solution
(a) Linearly dependent if and only if
d
= 10 as
1
3

1

1

5
5
4
5
d
row reduces to
1
0
5
0
1

2
0
0

10 +
d
.
(b) Linearly dependent if and only if
d
=

14 as
3

6
9
7
d
1

2
4

4
row reduces to
1

2
0
0
d
+ 14
0
0
0
1
.
10. Determine, with reasons, whether the following statements are (A) always true, (B) always false,
or (C) might be true or false.
(a) If
v
1
, ...,
v
5
∈
R
5
and
v
1
=
v
2
+
v
3
, then the set
{
v
1
, . . . ,
v
5
}
is linearly dependent.
(b) If
v
1
and
v
2
∈
R
2
lie on the same straight line through the origin, then the set
{
v
1
,
v
2
}
is
linearly dependent.